Some Energy Properties of Yang-Mills Connections

TL;DR

This paper investigates energy properties of Yang-Mills connections on vector bundles over compact Riemannian manifolds, establishing inequalities and energy bounds that reveal concentration phenomena and conditions for flatness.

Contribution

It proves a mean value inequality for the curvature density and demonstrates an energy concentration principle, providing new insights into the energy behavior of Yang-Mills connections.

Findings

Established a mean value inequality for |F_A|^{n/2}

Proved an energy concentration principle for bounded energy sequences

Showed energy is bounded below unless the bundle is flat

Abstract

We consider a vector bundle $E$ over a compact Riemannian manifold $M$=$M^{n}$,$n\geq 4$,and $A$ is a Yang-Mills connection with $L^{\frac{n}{2}}$ curvature $F_{A}$ on $E$.Then we prove a mean value inequality for the density $|F_{A}|^{\frac{n}{2}}$.This inequality give rise to an energy concentrate principle for sequences of solutions that have bounded energy.We also proof that the energy must be bounded from below by some positive constant unless $E$ is a flat bundle.